Tower design

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar
Indian Institute of Technology Madras
7.4 Tower Design
Once the external loads acting on the tower are determined, one proceeds
with an analysis of the forces in various members with a view to fixing up their
sizes. Since axial force is the only force for a truss element, the member has to
be designed for either compression or tension. When there are multiple load
conditions, certain members may be subjected to both compressive and tensile
forces under different loading conditions. Reversal of loads may also induce
alternate nature of forces; hence these members are to be designed for both
compression and tension. The total force acting on any individual member under
the normal condition and also under the broken- wire condition is multiplied by
the corresponding factor of safety, and it is ensured that the values are within the
permissible ultimate strength of the particular steel used.
Bracing systems
Once the width of the tower at the top and also the level at which the
batter should start are determined, the next step is to select the system of
bracings. The following bracing systems are usually adopted for transmission line
towers.
Single web system (Figure 7.29a)
It comprises either diagonals and struts or all diagonals. This system is
particularly used for narrow-based towers, in cross-arm girders and for portal
type of towers. Except for 66 kV single circuit towers, this system has little
application for wide-based towers at higher voltages.

Double web or Warren system (Figure 7.29b)
This system is made up of diagonal cross bracings. Shear is equally
distributed between the two diagonals, one in compression and the other in
tension. Both the diagonals are designed for compression and tension in order to
permit reversal of externally applied shear. The diagonal braces are connected at
their cross points. Since the shear perface is carried by two members and critical
length is approximately half that of a corresponding single web system. This
system is used for both large and small towers and can be economically adopted
throughout the shaft except in the lower one or two panels, where diamond or
portal system of bracings is more suitable.
Pratt system (Figure 7.29c)
This system also contains diagonal cross bracings and, in addition, it has
horizontal struts. These struts are subjected to compression and the shear is
taken entirely by one diagonal in tension, the other diagonal acting like a
redundant member.
It is often economical to use the Pratt bracings for the bottom two or three
panels and Warren bracings for the rest of the tower.
Portal system (Figure 7.29d)
The diagonals are necessarily designed for both tension and compression
and, therefore, this arrangement provides more stiffness than the Pratt system.
The advantage of this system is that the horizontal struts are supported at mid
length by the diagonals.

Like the Pratt system, this arrangement is also used for the bottom two or
three panels in conjuction with the Warren system for the other panels. It is
specially useful for heavy river-crossing towers.
Where
p = longitudinal spacing (stagger), that is, the distance between two
successive holes in the line of holes under consideration,
g = transverse spacing (gauge), that is, the distance between the same two
consecutive holes as for p, and
d = diameter of holes.
For holes in opposite legs of angles, the value of 'g' should be the sum of the
gauges from the back of the angle less the thickness of the angle.
Figure 7.29 Bracing syatems

Net effective area for angle sections in tension
In the case of single angles in tension connected by one leg only, the net
effective section of the angle is taken as
Aeff = A + Bk (7.28)
Where
A = net sectional area of the connected leg,
B = area of the outstanding leg = (l -t)t,
l = length of the outstanding leg,
t = thickness of the leg, and
1
k
B
1 0.35
A
=
+
In the case of a pair of angles back to back in tension connected by only
one leg of each angle to the same side of the gusset,
1
k
B
1 0.2
A
=
+
The slenderness ratio of a member carrying axial tension is limited to 375.
7.4.1 Compression members
While in tension members, the strains and displacements of stressed
material are small, in members subjected to compression, there may develop
relatively large deformations perpendicular to the centre line, under certain
criticallol1ding conditions.

The lateral deflection of a long column when subjected to direct load is
known as buckling. A long column subjected to a small load is in a state of stable
equilibrium. If it is displaced slightly by lateral forces, it regains its original
position on the removal of the force. When the axial load P on the column
reaches a certain critical value Pcr, the column is in a state of neutral equilibrium.
When it is displaced slightly from its original position, it remains in the displaced
position. If the force P exceeds the critical load Pcr, the column reaches an
unstable equilibrium. Under these circum- stances, the column either fails or
undergoes large lateral deflections.
Table 7.30 Effective slenderness ratios for members with different end
restraint
Type of member KL / r
a) Leg sections or joint members bolted at connections in both faces. L/r
b) Members with eccentric loading at both ends of the unsupported panel
with value of L / r up to and including 120
L/r
c) Members with eccentric loading at one end and normal eccentricities at
the other end of unsupported panel with values of L/r up to and including 120
30+0.75 L/r
d) Members with normal framing eccentricities at both ends of the
unsupported panel for values of L/r up to and including 120
60+0.5 L/r
e) Members unrestrained against rotation at both end of the unsupported
panel for values of L/r from 120 to 200.
L/r
f) Members partially restrained against rotation at one end of the
unsupported panel for values of L/r over 120 but up to and including 225
28.6+0.762 L/r
g) members partially restrained against rotation at both ends of unsupported
panel for values of L/r over 120 up to and including 250
46.2+0.615 L/r
Slenderness ratio
In long columns, the effect of bending should be considered while
designing. The resistance of any member to bending is governed by its flexural
rigidity EI where I =Ar2. Every structural member will have two principal moments
of inertia, maximum and minimum. The strut will buckle in the direction governed
by the minimum moment of inertia. Thus,

Imin = Armin
2
(7.29)
Where rmin is the least radius of gyration. The ratio of effective length of
member to the appropriate radius of gyration is known as the slenderness ratio.
Normally, in the design procedure, the slenderness ratios for the truss elements
are limited to a maximum value.
IS: 802 (Part 1)-1977 specifies the following limiting values of the
slenderness ratio for the design of transmission towers:
Leg members and main members in the cross-arm in compression 150
Members carrying computed stresses 200
Redundant members and those carrying nominal stresses 250
Tension members 350
Effective length
The effective length of the member is governed by the fixity condition at
the two ends.
The effective length is defined as 'KL' where L is the length from centre to
centre of intersection at each end of the member, with reference to given axis,
and K is a non-dimensional factor which accounts for different fixity conditions at
the ends, and hence may be called the restraint factor. The effective slenderness
ratio KL/r of any unbraced segment of the member of length L is given in Table
7.30, which is in accordance with 18:802 (Part 1)-1977.

Figure 7.30 Nomogram showing the variation of the effective slenderness
ratio kl / rL / r and the corresponding unit stress
Figure 7.30 shows the variation of effective slenderness ratio KL / r with L
/ r of the member for the different cases of end restraint for leg and bracing
members.
The value of KL / r to be chosen for estimating the unit stress on the
compression strut depends on the following factors:
1. the type of bolted connection
2. the length of the member
3. the number of bolts used for the connection, i.e., whether it is a
single-bolted or multiple-bolted connection
4. the effective radius of gyration

Table 7.31 shows the identification of cases mentioned in Table 7.30 and
Figure 7.30 for leg and bracing members normally adopted. Eight different cases
of bracing systems are discussed in Table 7.31.
SI.
No
1
Member
2
Method of
loading
3
Rigidity of joint
4
L/r ratio
5
Limiting
values
of L/r
6
Categorisation
of member
7
KL/r
8
0 to
120
Case (a) L/r
1 Concentric No restraint at ends L/rw
120 to
150
Case (e) L/r
L/rxx or
L/ryy or
0.5L/rw
0 to
120
Case (a) L/r
2 Concentric No restraint at ends
L/rxx or
L/ryy or
0.5L/rw
120 to
150
Case (e) L/r
unsupported panel-no
restraint at ends
L/rw
0 to
120
Case (d)
60
+0.5L/r
L/rw
120 to
200
Case (e) L/r
3 eccentric
L/rw
120 to
250
Case (g)
46.2 +
0.615L/r
4 concentric No restraint at ends
max of
L/rxx or
L/ryy
0 to
120
Case (b) L/r

max of
L/rxx or
L/ryy
120 to
200
Case (e) L/r
max of
L/rxx or
L/ryy
120 to
250
Case (g)
46.2 +
0.615L/r
concentric
at ends and
eccentric at
intermediate
joints in
both
directions
0.5L/ryy
or L/rxx
0 to
120
Case (e)
30 +
0.75L/r
concentric
at ends and
intermediate
joints
0.5L/ryy
or L/rxx
0 to
120
Case (a) L/r
5
concentric
at ends
Multiple bolt
connections Partial
restraints at ends and
intermediate joints
0.5L/ryy
or L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r
Single bolt
No restraint at ends
0.5L/rw
or
0.75L/rxx
0 to
120
Case (c)
30 +
0.75L/reccentric
(single
angle)
Single bolt
No restraint at ends
0.5L/rw
or
0.75L/rxx
120 to
200
Case (e) L/r
6
concentric
(Twin angle)
Multiple bolt
connections
Partial restraints at
ends
and intermediate joints
0.5L/rw
or
0.75L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r
7
eccentric
(single
angle)
Single or multiple bolt
connection
0.5L/rw
or L/rxx
0 to
120
Case (g)
60 +
0.5L/r

Table 7.31 Categorisation of members according to eccentricity of loading
and end restraint conditions
Euler failure load
Euler determined the failure load for a perfect strut of uniform cross-
section with hinged ends. The critical buckling load for this strut is given by:
2 2
cr 2 2
EI EA
P
L L
r
π π
= =
⎛ ⎞
⎜ ⎟
⎝ ⎠
(7.30)
Single bolt connection,
no restraint at ends
and at intermediate
joints.
0.5L/rw
or L/rxx
120 to
200
Case (e) L/r
Multiple bolt at ends
and single bolt at
intermediate joints
0.5L/rw
120 to
225
Case (f)
28.6 +
.762L/r
and at intermediate
joints Partial restraints
at both ends
L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r
ends and at
intermediate joints
0.5L/rw
or L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r
Single or multiple bolt
connection
0.5L/ryy
or L/rxx
0 to
120
Case (a) L/r
Single bolt connection,
no restraint at ends
and at intermediate
joints.
0.5L/ryy
or L/rxx
120 to
200
Case (e) L/r
and single bolt at
intermediate joints
0.5L/ryy
120 to
200
Case (f)
28.6 +
.762L/r
Multiple bolt
connection Partial
restraints at both ends
L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r
8
eccentric
(single
angle)
ends and at
intermediate joints
0.5L/ryy
or L/rxx
120 to
250
Case (g)
46.2 +
0.615L/r

The effective length for a strut with hinged ends is L.
At values less than π2
EL/L2
the strut is in a stable equilibrium. At values of
P greater than π2
EL/L2
the strut is in a condition of unstable equilibrium and any
small disturbance produces final collapse. This is, however, a hypothetical
situation because all struts have some initial imperfections and thus the load on
the strut can never exceed π2
EL/L2
. If the thrust P is plotted against the lateral
displacement ∆ at any section, the P - ∆ relationship for a perfect strut will be as
shown in Figure 7.31 (a).
In this figure, the lateral deflections occurring after reaching critical
buckling load are shown, that is 2 2
crP EI / L≥ π , When the strut has small
imperfections, displacement is possible for all values of P and the condition of
neutral equilibrium P = π2
EL/L2
is never attained. All materials have a limit of
proportionality. When this is reached, the flexural stiffness decreases initiating
failure before P = π2
EL/L2
is reached (Figure 7.31 (b))
Empirical formulae
The following parameters influence the safe compressive stress on the
column:
1. Yield stress of material
2. Initial imperfectness
3. (L/r) ratio
4. Factor of safety
5. End fixity condition
6. (b/t) ratio (Figure 7.31)) which controls flange buckling

Figure 7.31 (d) shows a practical application of a twin-angle strut used in a
typical bracing system.
Taking these parameters into consideration, the following empirical
formulae have been used by different authorities for estimating the safe
compressive stress on struts:
1. Straight line formula
2. Parabolic formula
3. Rankine formula
4. Secant or Perry's formula
These formulae have been modified and used in the codes evolved in
different countries.
IS: 802 (Part I) -1977 gives the following formulae which take into account
all the parameters listed earlier.
For the case b / t ≤ 13 (Figure 7.30 (c)),
2
2
a
KL
r
F 2600 kg / cm
12
⎧ ⎫⎛ ⎞
⎪ ⎪⎜ ⎟
⎪ ⎪⎝ ⎠= −⎨ ⎬
⎪ ⎪
⎪ ⎪
⎩ ⎭
(7.31)
Where KL / r ≤ 120
6
2
a 2
20x10
F kg / cm
KL
r
=
⎛ ⎞
⎜ ⎟
⎝ ⎠
(7.32)

Where KL / r > 120
Fcr = 4680 - 160(b / t) kg/cm2
Where 13 < b/t < 20 (7.33)
2
cr 2
590000
F kg / cm
b
t
=
⎛ ⎞
⎜ ⎟
⎝ ⎠
Where b / t > 20 (7.34)
Where
Fa = buckling unit stress in compression,
Fcr = limiting crippling stress because of large value of b / t,
b = distance from the edge of fillet to the extreme fibre, and
t = thickness of material.
Equations (7.31) and (7.32) indicate the failure load when the member
buckles and Equations (7.33) and (7.34) indicate the failure load when the flange
of the member fails.
Figure 7.30 gives the strut formula for the steel with a yield stress of 2600
kg/sq.cm. with respect to member failure. The upper portion of the figure shows
the variation of unit stress with KL/r and the lower portion variation of KL/r with
L/r. This figure can be used as a nomogram for estimating the allowable stress
on a compression member.

An example illustrating the procedure for determining the effective length,
the corresponding slenderness ratio, the permissible unit stress and the
compressive force for a member in a tower is given below.
Figure 7.31

Example
Figure 7.31 (d) shows a twin angle bracing system used for the horizontal
member of length L = 8 m. In order to reduce the effective length of member AB,
single angle CD has been connected to the system. AB is made of two angles
100 x 100mm whose properties are given below:
rxx = 4.38 cm
ryy = 3.05 cm
Area = 38.06 sq.cm.
Double bolt connections are made at A, Band C. Hence it can be assumed
that the joints are partially restrained. The system adopted is given at SL. No.8 in
Table 7.31. For partial restraint at A, B and C,
L/r = 0.5 L/ryy or L/rxx
= 0.5 x 800/3.05 or 800/4.38
= 131.14 or 182.64
The governing value of L/r is therefore182.64, which is the larger of the
two values obtained. This value corresponds to case (g) for which KL/r
= 46.2 + 0.615L/r
= 158.52
Note that the value of KL/r from the curve is also 158.52 (Figure 7.30).
The corresponding stress from the curve above is 795 kg/cm2
, which is shown
dotted in the nomogram. The value of unit stress can also be calculated from
equation (7.32). Thus,

6
2
a 2
20x10
F kg / cm
KL
r
=
⎛ ⎞
⎜ ⎟
⎝ ⎠
= 20 x 106
/ 158.52 x 158.52
= 795 kg/cm2
.
The safe compression load on the strut AB is therefore
F = 38.06 x 795
= 30,257 kg
7.4.2 Computer-aided design
Two computer-aided design methods are in vogue, depending on the
computer memory. The first method uses a fixed geometry (configuration) and
minimizes the weight of the tower, while the second method assumes the
geometry as unknown and derives the minimization of weight.
Method 1: Minimum weight design with assumed geometry
Power transmission towers are highly indeterminate and are subjected to
a variety of loading conditions such as cyclones, earthquakes and temperature
variations.
The advent of computers has resulted in more rational and realistic
methods of structural design of transmission towers. Recent advances in
optimisation in structural design have also been incorporated into the design of
such towers.
While choosing the member sizes, the large number of structural
connections in three dimensions should be kept in mind. The selection of

members is influenced by their position in relation to the other members and the
end connection conditions. The leg sections which carry different stresses at
each panel may be assigned different sizes at various levels; but consideration of
the large number of splices involved indicates that it is usually more economical
and convenient, even though heavier, to use the same section for a number of
panels. Similarly, for other members, it may be economical to choose a section of
relatively large flange width so as to eliminate gusset plates and correspondingly
reduce the number of bolts.
In the selection of structural members, the designer is guided by his past
experience gained from the behavior of towers tested in the test station or
actually in service. At certain critical locations, the structural members are
provided with a higher margin of safety, one example being the horizontal
members where the slope of the tower changes and the web members of panels
are immediately below the neckline.
Optimisation
Many designs are possible to satisfy the functional requirements and a
trial and error procedure may be employed to choose the optimal design.
Selection of the best geometry of a tower or the member sizes is examples of
optimal design procedures. The computer is best suited for finding the optimal
solutions. Optimisation then becomes an automated design procedure, providing
the optimal values for certain design quantities while considering the design
criteria and constraints.
Computer-aided design involving user-ma- chine interaction and
automated optimal design, characterized by pre-programmed logical decisions,

based upon internally stored information, are not mutually exclusive, but
complement each other. As the techniques of interactive computer-aided design
develop, the need to employ standard routines for automated design of structural
subsystems will become increasingly relevant.
The numerical methods of structure optimisation, with application of
computers, automatically generate a near optimal design in an iterative manner.
A finite number of variables has to be established, together with the constraints,
relating to these variables. An initial guess-solution is used as the starting point
for a systematic search for better designs and the process of search is
terminated when certain criteria are satisfied.
Those quantities defining a structural system that are fixed during the
automated design are- called pre-assigned parameters or simply parameters and
those quantities that are not pre-assigned are called design variables. The
design variables cover the material properties, the topology of the structure, its
geometry and the member sizes. The assignment of the parameters as well as
the definition of their values is made by the designer, based on his experience.
Any set of values for the design variables constitutes a design of the
structure. Some designs may be feasible while others are not. The restrictions
that must be satisfied in order to produce a feasible design are called constraints.
There are two kinds of constraints: design constraints and behavior constraints.
Examples of design constraints are minimum thickness of a member, maximum
height of a structure, etc. Limitations on the maximum stresses, displacements or
buckling strength are typical examples of behavior constraints. These
constraints are expressed mathematically as a set of inequalities:

{ }( )jg X 0 j 1,2,....,m≤ = (7.35a)
Where {X} is the design vector, and
m is the number of inequality constraints.
In addition, we have also to consider equality constraints of the form
{ }( )jh X 0 j 1,2,....,k≤ = (7.35b)
Where k is the number of equality constraints.
Example
The three bar truss example first solved by Schmit is shown in Figure
7.32. The applied loadings and the displacement directions are also shown in this
figure.
Figure 7.32 Two dimensional plot of the design variables X1 and X2
1. Design constraints: The condition that the area of members cannot be
less than zero can be expressed as

1 1
2 2
g X 0
g X 0
≡ − ≤
≡ − ≤
2. Behaviour constraints: The three members of the truss should be safe,
that is, the stresses in them should be less than the allowable stresses in tension
(2,000 kg/cm2
) and compression (1,500kg/cm2
). This is expressed as
3 1g 2,000 0≡ σ − ≤ Tensile stress limitation in member 1
4 1g 1,500 0≡ −σ − ≤
5 2
6 2
7 3
8 3
g 2,000 0
g 1,500 0
g 2,000 0
g 1,500 0
≡ σ − ≤
≡ −σ − ≤
≡ σ − ≤
≡ −σ − ≤
Compressive stress limitation in member 2 and so on
3. Stress force relationships: Using the stress-strain relationship σ = [E] {∆}
and the force-displacement relationship F = [K] {∆}, the stress-force relationship
is obtained as {s} = [E] [K]-1
[F] which can be shown as
2 1
1 2
1 2 1
1
2 2
1 2 1
2
1 2
1 2 1
X 2X
2000
2X X 2X
2X
2000
2X X 2X
X
2000
2X X 2X
⎛ ⎞+
σ = ⎜ ⎟⎜ ⎟+⎝ ⎠
⎛ ⎞
σ = ⎜ ⎟⎜ ⎟+⎝ ⎠
⎛ ⎞
σ = ⎜ ⎟⎜ ⎟+⎝ ⎠
4. Constraint design inequalities: Only constraints g3, g5, g8 will affect the
design. Since these constraints can now be expressed in terms of design
variables X1 and X2 using the stress force relationships derived above, they can

be represented as the area on one side of the straight line shown in the two-
dimensional plot (Figure 7.32 (b)).
Design space
Each design variable X1, X2 ...is viewed as one- dimension in a design
space and a particular set of variables as a point in this space. In the general
case of n variables, we have an n-dimensioned space. In the example where we
have only two variables, the space reduces to a plane figure shown in Figure
7.32 (b). The arrows indicate the inequality representation and the shaded zone
shows the feasible region. A design falling in the feasible region is an
unconstrained design and the one falling on the boundary is a constrained
design.
Objective function
An infinite number of feasible designs are possible. In order to find the
best one, it is necessary to form a function of the variables to use for comparison
of feasible design alternatives. The objective (merit) function is a function whose
least value is sought in an optimisation procedure. In other words, the
optimization problem consists in the determination of the vector of variables X
that will minimise a certain given objective function:
Z = F ({X}) 7.35(c)
In the example chosen, assuming the volume of material as the objective
function, we get
Z = 2(141 X1) + 100 X2

The locus of all points satisfying F ({X}) = constant, forms a straight line in
a two-dimensional space. In this general case of n-dimensional space, it will form
a surface. For each value of constraint, a different straight line is obtained. Figure
7.32 (b) shows the objective function contours. Every design on a particular
contour has the same volume or weight. It can be seen that the minimum value
of F ( {X} ) in the feasible region occurs at point A.

Figure 7.33 Configuration and loading condition for the example tower
There are different approaches to this problem, which constitute the
various methods of optimization. The traditional approach searches the solution
by pre-selecting a set of critical constraints and reducing the problem to a set of
equations in fewer variables. Successive reanalysis of the structure for improved
sets of constraints will tend towards the solution. Different re-analysis methods

can be used, the iterative methods being the most attractive in the case of
towers.
Optimality criteria
An interesting approach in optimization is a process known as optimality
criteria. The approach to the optimum is based on the assumption that some
characteristics will be attained at such optimum. The well-known example is the
fully stressed design where it is assumed that, in an optimal structure, each
member is subjected to its limiting stress under at least one loading condition.
The optimality criteria procedures are useful for transmission lines and
towers because they constitute an adequate compromise to obtain practical and
efficient solutions. In many studies, it has been found that the shape of the
objective function around the optimum is flat, which means that an experienced
designer can reach solutions, which are close to the theoretical optimum.
Mathematical programming
It is difficult to anticipate which of the constraints will be critical at the
optimum. Therefore, the use of inequality constraints is essential for a proper
formulation of the optimal design problem.
The mathematical programming (MP) methods are intended to solve the
general optimisation problem by numerical search algorithms while being general
regarding the objective function and constraints. On the other hand,
approximations are often required to be efficient on large practical problems such
as tower optimisation.

Optimal design processes involve the minimization of weight subject to
certain constraints. Mathematical programming methods and structural theorems
are available to achieve such a design goal.
Of the various mathematical programming methods available for
optimisation, the linear programming method is widely adopted in structural
engineering practice because of its simplicity. The objective function, which is the
minimisation of weight, is linear and a set of constraints, which can be expressed
by linear equations involving the unknowns (area, moment of inertia, etc. of the
members), are used for solving the problems. This can be mathematically
expressed as follows.
Suppose it is required to find a specified number of design variables x1,
x2.....xn such that the objective function
Z = C1 x1 + C2 x2 + ....Cn xn
is minimised, satisfying the constraints
11 1 12 2 1n n 1
21 1 22 2 2n n 2
m1 1 m2 2 mn n m
a x a x ..........a x b
a x a x ..........a x b
.
.
.
a x a x ..........a x b
+ + ≤
+ + ≤
+ + ≤
(7.36)
The simplex algorithm is a versatile procedure for solving linear
programming (LP) problems with a large number of variables and constraints.

The simplex algorithm is now available in the form of a standard computer
software package, which uses the matrix representation of the variables and
constraints, especially when their number is very large.
The equation (7.36) is expressed in the matrix form as follows:
Find
1
2
n
x
x
X
x
⎧ ⎫
⎪ ⎪
⎪ ⎪⎪ ⎪
= −⎨ ⎬
⎪ ⎪−
⎪ ⎪
⎪ ⎪⎩ ⎭
which minimises the objective function
( )
n
i i
i 1
f x C x
−
= ∑ (7.37)
subject to the constraints,
n
jk k j
k 1
i
a x b , j 1,2,...m
andx 0, i 1,2,...n
−
= =
≥ =
∑ (7.38)
where Ci, ajk and bj are constants.
The stiffness method of analysis is adopted and the optimisation is
achieved by mathematical programming.
The structure is divided into a number of groups and the analysis is
carried out group wise. Then the member forces are determined. The critical
members are found out from each group. From the initial design, the objective
function and the constraints are framed. Then, by adopting the fully stressed

design (optimality criteria) method, the linear programming problem is solved and
the optimal solution found out. In each group, every member is designed for the
fully stressed condition and the maximum size required is assigned for all the
members in that group. After completion of the design, one more analysis and
design routine for the structure as a whole is completed for alternative cross-
sections.
Example
A 220 k V double circuit tangent tower is chosen for study. The basic
structure, section plan at various levels and the loading conditions are tentatively
fixed. The number of panels in the basic determinate structure is 15 and the
number of members is 238. Twenty standard sections have been chosen in the
increasing order of weight. The members have been divided into eighteen
groups, such as leg groups, diagonal groups and horizontal groups, based on
various panels of the tower. For each group a section is specified.
Normal loading conditions and three broken- wire conditions has been
considered. From the vertical and horizontal lengths of each panel, the lengths of
the members are calculated and the geometry is fixed. For the given loading
conditions, the forces in the various members are computed, from which the
actual stresses are found. These are compared with allowable stresses and the
most stressed member (critical) is found out for each group. Thereafter, an initial
design is evolved as a fully stressed design in which critical members are
stressed up to an allowable limit. This is given as the initial solution to simplex
method, from which the objective function, namely, the weight of the tower, is
formed. The initial solution so obtained is sequentially improved, subject to the
constraints, till the optimal solution is obtained.

In the given solution, steel structural angles of weights ranging from 5.8
kg/m to 27.20 kg/m are utilised. On the basis of the fully stressed design,
structural sections of 3.4 kg/m to 23.4 kg/m are indicated and the corresponding
weight is 5,398 kg. After the optimal solution, the weight of the tower is 4,956 kg,
resulting in a saving of about 8.1 percent.
Method 2: Minimum weight design with geometry as variable
In Method 1, only the member sizes were treated as variables whereas
the geometry was assumed as fixed. Method 2 treats the geometry also as a
variable and gets the most preferred geometry. The geometry developed by the
computer results in the minimum weight of tower for any practically acceptable
configuration. For solution, since an iterative procedure is adopted for the
optimum structural design, it is obvious that the use of a computer is essential.
The algorithm used for optimum structural design is similar to that given by
Samuel L. Lipson which presumes that an initial feasible configuration is
available for the structure. The structure is divided into a number of groups and
the externally applied loadings are obtained. For the given configuration, the
upper limits and the lower limits on the design variables, namely, the joint
coordinates are fixed. Then (k-1) new configurations are generated randomly as
xij = li + rij( ui - li ) (7.39)
i = 1, 2 ...n
j = 1, 2 ...k
where k is the total number of configurations in the complex, usually larger
than (n + 1), where n is the number of design variables and rij is the random
number for the ith
coordinate of the jth
point, the random numbers having a

uniform distribution over the interval 0 to 1 and ui is the upper limit and Li is the
lower limit of the ith
independent variable.
Thus, the complex containing k number of feasible solutions is generated
and all these configurations will satisfy the explicit constraints, namely, the upper
and lower bounds on the design variables. Next, for all these k configurations,
analysis and fully stressed designs are carried out and their corresponding total
weights determined. Since the fully stressed design concept is an eco nomical
and practical design, it is used for steel area optimisation. Every area
optimisation problem is associated with more than one analysis and design. For
the analysis of the truss, the matrix method described in the previous chapter has
been used. Therefore, all the generated configurations also satisfy the implicit
constraints, namely, the allowable stress constraints.
From the value of the objective function (total weight of the structure) of k
configurations, the vector, which yields the maximum weight, is searched and
discarded, and the centroid c of each joint of the k-1 configurations is determined
from
( )ic ij iw
j 1
1
x K x x
K 1 −
⎧ ⎫⎪ ⎪
= −⎨ ⎬
− ⎪ ⎪⎩ ⎭
∑ (7.40)
i = 1, 2, 3 ... n
in which xic and xiw are the ith
coordinates of the centroid c and the discarded
point w.
Then a new point is generated by reflecting the worst point through the
centroid, xic

That is, xiw = xic + α ( xic - xiw )
(7.41)
i = 1,2,..... n where α is a constant.
Figure 7.34 Node numbers
This new point is first examined to satisfy the explicit constraints. If it
exceeds the upper or lower bound value, then the value is taken as the
corresponding limiting value, namely, the upper or lower bound. Now the area
optimisation is carried out for the newly generated configuration and the
functional value (weight) is determined. If this functional value is better than the
second worst, the point is accepted as an improvement and the process of
developing the new configuration is repeated as mentioned earlier. Otherwise,
the newly generated point is moved halfway towards the centroid of the
remaining points and the area optimisation is repeated for the new configuration.

This process is repeated over a fixed number of iterations and at the end of every
iteration, the weight and the corresponding configuration are printed out, which
will show the minimum weight achievable within the limits (l and u) of the
configuration.
Example
The example chosen for the optimum structural design is a 220 k V
double-circuit angle tower. The tower supports one ground wire and two circuits
containing three conductors each, in vertical configuration, and the total height of
the tower is 33.6 metres. The various load conditions are shown in Figure 7.33.
The bracing patterns adopted are Pratt system and Diamond system in
the portions above and below the bottom-most conductor respectively. The initial
feasible configuration is shown on the top left corner of Figure 7.33. Except x, y
and z coordinates of the conductor and the z coordinates of the foundation
points, all the other joint coordinates are treated as design variables. The tower
configuration considered in this example is restricted to a square type in the plan
view, thus reducing the number of design variables to 25.
In the initial complex, 27 configurations are generated, including the initial
feasible configuration. Random numbers required for the generation of these
configurations are fed into the comJ7llter as input. One set containing 26 random
numbers with uniform distribution over the interval 0 to 1 are supplied for each
design variable. Figure 7.34 and Figure 7.35 show the node numbers and
member numbers respectively.

The example contains 25 design variables, namely, the x and y
coordinates of the nodes, except the conductor support points and the z
coordinates of the support nodes (foundations) of the tower. 25 different sets of
random numbers, each set containing 26 numbers, are read for 25 design
variables. An initial set of27 configuration is generated and the number of
iterations for the development process is restricted to 30. The weight of the tower
for the various configurations developed during optimisation procedure is
pictorially represented in Figure 7.36. The final configuration is shown in Figure
7.37a and the corresponding tower weight, including secondary bracings, is
5,648 kg.
Figure 7.35 Member numbers

Figure 7.36 Tower weights for various configurations generated
Figure 7.37

Figure 7.38 Variation of tower weight with base width
Figure 7.39 Tower geometry describing key joints and joints obtained from
key joints
This weight can further be reduced by adopting the configuration now
obtained as the initial configuration and repeating the search by varying the
controlling coordinates x and z. For instance, in the present example, by varying
the x coordinate, the tower weight has been reduced to 5,345 kg and the

corresponding configuration is shown in Figure 7.37b. Figure 7.38 shows the
variation of tower weight with base width.
In conclusion, the probabilistic evaluation of loads and load combinations
on transmission lines, and the consideration of the line as a whole with towers,
foundations, conductors and hardware, forming interdependent elements of the
total system with different levels of safety to ensure a preferred sequence of
failure, are all directed towards achieving rational behaviour under various
uncertainties at minimum transmission line cost. Such a study may be treated as
a global optimisation of the line cost, which could also include an examination of
alternative uses of various types of towers in a family, materials to be employed
and the limits to which different towers are utilised as discrete variables and the
objective function as the overall cost.

7.4.3 Computer software packages
Figure 7.40 Flowchart for the development of tower geometry in the
OPSTAR program
The general practice is to fix the geometry of the tower and then arrive at
the loads for design purposes based on which the member sizes are determined.
This practice, however, suffers from the following disadvantages:
1. The tower weight finally arrived at may be different from the assumed
design weight.
2. The wind load on tower calculated using assumed sections may not
strictly correspond to the actual loads arrived at on the final sections adopted.
3. The geometry assumed may not result in the economical weight of tower.

4. The calculation of wind load on the tower members is a tedious process.
Most of the computer software packages available today do not enable the
designer to overcome the above drawbacks since they are meant essentially to
analyse member forces.
Figure 7.41 Flowchart for the solution sequence (opstar programme)
In Electricite de France (EDF), the OPSTAR program has been used for
developing economical and reliable tower designs. The OPSTAR program
optimises the tower member sizes for a fixed configuration and also facilitates the

development of new configurations (tower outlines), which will lead to the
minimum weight of towers. The salient features of the program are given below:
Geometry: The geometry of the tower is described by the coordinates of the
nodes. Only the coordinates of the key nodes (8 for a tower in Figure 7.39)
constitute the input. The computer generates the other coordinates, making use
of symmetry as well as interpolation of the coordinates of the nodes between the
key nodes. This simplifies and minimises data input and aids in avoiding data
input errors.
Solution technique: A stiffness matrix approach is used and iterative
analysis is performed for optimisation.
Description of the program: The first part of the program develops the
geometry (coordinates) based on data input. It also checks the stability of the
nodes and corrects the unstable nodes. The flow chart for this part is given in
Figure 7.40.
The second part of the program deals with the major part of the solution
process. The input data are: the list of member sections from tables in
handbooks and is based on availability; the loading conditions; and the boundary
conditions.
The solution sequence is shown in Figure 7.41. The program is capable of
being used for either checking a tower for safety or for developing a new tower
design. The output from the program includes tower configuration; member sizes;
weight of tower; foundation reactions under all loading conditions; displacement

of joints under all loading conditions; and forces in all members for all loading
conditions.
7.4.4 Tower accessories
Designs of important tower accessories like Hanger, Step bolt, Strain
plate; U-bolt and D-shackle are covered in this section. The cost of these tower
accessories is only a very small fraction of the S overall tower cost, but their
failure will render the tower functionally ineffective. Moreover, the towers have
many redundant members whereas the accessories are completely determinate.
These accessories will not allow any load redistribution, thus making failure
imminent when they are overloaded. Therefore, it is preferable to have larger
factors of safety associated with the tower accessories than those applicable to
towers.
Hanger (Figure 7.42)

Figure 7.42 Hanger
The loadings coming on a hanger of a typical 132 kV double-circuit tower
are given below:
Type of loading NC BWC
Transverse 480kg 250kg
Vertical 590kg 500kg
Longitudinal - 2,475kg
Maximum loadings on the hanger will be in the broken-wire condition and
the worst loaded member is the vertical member.
Diameter of the hanger leg = 21mm
Area = p x (21)2
/ 4 x 100 = 3.465 sq.cm.
Maximum allowable tensile stress for the steel used = 3,600 kg/cm2
Allowable load = 3,600 x 3.465
= 12,474 kg.
Dimensions
Nom
bolt
dia
threads Shank
dia
ds
Head
dia
dk
Head
thickness
k
Neck
radius
(app)
r
Bolt
length
l
Thread
length
b
Width
across
flats
s
Nut thickness
m
Metric Serious (dimensions in mm before galvanising)
+1.10 +2 +1 +3 +5 +0
16 m 16 16
-0.43
35
-0
6
-0
3 175
-0
60
-0
24
-0.84
13 0.55

Figure 7.43 Dimensions and mechanical properties of step bolts and nuts
Loads in the vertical leg
1. Transverse load (BWC) = 250 / 222 x 396
= 446kg.
2. Longitudinal load = 2,475 kg.
3. Vertical load = 500 kg.
Total = 3,421kg.
It is unlikely that all the three loads will add up to produce the tension in
the vertical leg. 100 percent effect of the vertical load and components of
longitudinal and transverse load will be acting on the critical leg to produce
maximum force. In accordance with the concept of making the design
conservative, the design load has been assumed to be the sum of the three and
hence the total design load = 3,421 kg.
Factor of safety = 12,474 / 3,421 = 3.65 which is greater than 2, and
hence safe
Step bolt (Figure 7.43)
Special mild steel hot dip galvanised bolts called step bolts with two
hexagonal nuts each, are used to gain access to the top of the tower structure.
The design considerations of such a step bolt are given below.
Bolts Nuts
1. Tensile strength - 400 N/mm2
min. 1. Proof load stress - 400 N/mm2
2. Brinell Hardness- HB 114/209
3. Cantilever load test - with 150kg
2. Brinell Hardness- HB 302 max

The total uniformly distributed load over the fixed length = 100 kg
(assumed).
The maximum bending moment
100 x 13 / 2 = 650 kg cm.
The moment of inertia = p x 164
/ 64 = 0.3218 cm4
Maximum bending stress = 650 x 0.8 / 0.3218
= 1,616 kg/cm2
Assuming critical strength of the high tensile steel = 3,600 kg/cm2
,
factor of safety = 3,600 / 1,616 = 2.23, which is greater than 2, and hence
safe.
Step bolts are subjected to cantilever load test to withstand the weight of
man (150kg).
Strain plate (Figure 7.44)
The typical loadings on a strain plate for a 132 kV double-circuit tower are
given below:
Vertical load = 725kg
Transverse load = 1,375kg
Longitudinal load = 3,300kg
Bending moment due to vertical load = 725 x 8 / 2 = 2,900kg.cm.
Ixx = 17 x (0.95)3
/ 12 = 1.2146 cm4
y (half the depth) = 0.475cm.

Figure 7.44 Strain plate
Section modulus Zxx = 1.2146 / 0.475 = 2.5568
Bending stress fxx = 2,900 / 2.5568 = 1,134 kg/cm2
Bending moment due to transverse load = 1.375 x 8 / 2 = 5,500 kg.cm.
Actually the component of the transverse load in a direction parallel to the
line of fixation should be taken into account, but it is safer to consider the full
transverse load.
Iyy = 0.95 x 173
/ 12 = 389 cm4
Zyy = 389 / 8.5 = 45.76
Bending stress fyy = 5500 / 45.76 = 120 kg/cm2
Total maximum bending stress
fxx + fyy = 1,134 + 120 = 1,254 kg/cm2
Direct stress due to longitudinal load = longitudinal load / Cross-sectional area
= 3,300 / 13.5 x 0.95
= 257.3 kg/cm2
Check for combined stress
The general case for a tie, subjected to bending and tension, is checked
using the following interaction relationship:

b t
b T
f f
1
F F
+ ≤ (7.36)
Where ft = actual axial tensile stress,
fb = actual bending tensile stress
Ft = permissible axial tensile stress, and
Fb = permissible bending tensile stress.
Assuming Ft = 1,400 kg/cm2
and Fb = 1,550 kg/cm2
. The expression
reduces to
= 1,254 / 1,550 + 257.3 / 1,400
= 0.9927 <1, hence safe.
Check for the plate in shear
Length of the plate edge under shear = 1.75 cm
Area under shear = 2 x 1.75 x 0.95
= 3.325 sq.cm.
Shearing stress =3,300 / 3,325 = 992 kg/cm2
Permissible shear stress = 1,000 kg/cm2
Hence, it is safe in shear.
Check for the plate in bearing
Pin diameter = 19mm
Bearing area = 1.9 x 0.95 = 1.805 cm2
Maximum tension in the conductor = 3,300 kg.
Bearing stress = = 1,828 kg/cm2
Permissible bearing stress = 1860 kg/cm2
Hence, it is safe in bearing.

Check for bolts in shear
Diameter of the bolt = 16mm
Area of the bolt = 2.01 sq.cm.
Shear stress = 3,300 / 3 x 2.01 = 549 kg/cm2
Permissible shearing stress = 1,000 kg/cm2
Hence, three 16mm diameter bolts are adequate.
U-bolt (Figure 7.45)
Figure 7.45 U-bolt
The loadings in a U-bolt for a typical 66 kV double circuit tower are given
below:
NC BWC
Transverse load = 216 108
Vertical load = 273 227
Longitudinal load = - 982
Permissible bending stress for mild steel = 1,500 kg/cm2
Permissible tensile stress = 1,400 kg/cm2
Let the diameter of the leg be 16mm.

The area of the leg = 2.01 sq.cm.
1. Direct stress due to vertical load = 273 / 2.01 x 2
= 67.91 kg/cm2
2. Bending due to transverse load (NC)
Bending moment = 216 x 5 = 1,080 kg.cm
Section Modulus = 2 x πd3
/ 32 = 2 x 3.14 x 1.63
/ 32
= 0.804
Bending stress = 1,080 / 0.804
= 1,343 kg/cm2
< 1,500 kg/cm2
Hence safe.
3. Bending due to longitudinal load (BWC)
Bending moment = 982 x 5 = 4,910 kg.cm
24 2
2 4
xx
d d
I x 2.5 25.77cm
64 4
⎛ ⎞π π
= + =⎜ ⎟⎜ ⎟
⎝ ⎠
y = 2.5 + 0.8
Bending stress = 4, 90 / 25.77 x (2.5 + 0.8) = 629 kg/cm2
In the broken-wire condition total bending stress = 1,343 / 2 + 629
= 1,300 kg/cm2
Hence, the worst loading will occur during normal condition.
For safe design,
b t
b t
f f
1
F F
+ ≤

67.91 / 1400 + 1343 / 1500 = 0.9365 < 1
Hence safe.
Bearing strength of the angle-bolt connection
Safe bearing stress for the steel used = 4,725 kg/cm2
Diameter of hole = 16mm + 1.5mm = 17.5mm
Thickness of the angle leg = 5mm
Under normal condition
Bearing stress = (216 + 273)/1.75 x 0.5 = 558.85 kg/cm2
Factor of safety = 4,725 / 558.85 = 8.45
Under broken-wire condition
Bearing stress = (108+227+982) / 1.75 x 0.5
= 1,505.14 kg/cm2
Therefore, factor of safety = 4,725 /1,505.14 = 3.13
Hence safe.

D-shackle (Figure 7.46)
Figure 7.46 D-Shackle
The loadings for a D-shackle for a 132 kV single circuit tower are given
below:
NC BWC
Transverse load 597 400
Vertical load 591 500
Longitudinal load - 1945
The D-shackle is made of high tensile steel. Assume permissible stress of
high tensile steel as 2,500kg/cm2
and 2,300kg/cm2
in tension and bearing respectively.

Normal condition
Area of one leg = π / 4 x (1.6) = 2.01 sq.cm.
Assuming the total load to be the sum of vertical and transverse loads
(conservative), the design load
= 597 + 591
= 1,188
Tensile stress = 1,188 / 2 x 1 / 2.01 = 295.5 kg/cm2
Factor of safety = 2,500 / 295.5 = 8.46
Shearing stress in the bolt =
2
597
x 2
4
π
= 190 kg/cm2
Factory of safety = 2,300 / 190 = 12.1
Broken-wire condition
Assuming the total load to be sum of the loads listed for broken-wire
condition,
Tensile stress in shackle = 2,845 / 2 x 1 / 2.01
= 707.711 kg/cm2
Factor of safety = 2,500 / 707.7 = 3.53
Shearing stress in the bolt =
( )2
2,845
2x x 2
4
π
= 452.8
Factor of safety = 2,300 / 452.8 = 5.07
Hence safe.

Tower design

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